I highly recommend you read (or at least skim), an essay by Paul Lockhart entitled A Mathematician’s Lament. I think it will generate interesting conversation, and I’d love to hear your comments. I have an opinion, but I’ll share it after I hear from some of you.
Random Walks
Mr. Chase blogs about math
Random Walks 
Does this have anything to do with Harry Potter? =\ Which you need to read, Mr. Chase. How difficult are mathmatical essays to comprehend?
Maybe over the weekend, Mr. Chase…
This is way too long.
Whew. I read like the first two pages. It was pretty interesting until it started talking about math… But I think that more people like Math, the way it is mandatorily taught now, than a paint by colors kit would have success.
Hello, this is Meredith, if you remember me from two years ago. One of your new students, Fiona, pointed me in the direction of your blog (and subsequently, the YouTube account).
First, you should get a prettier layout. Second, -bookmarks-. You made me all nostalgic again, because my 8th grade English teacher ran a blog for us and he was awesome and bald. He called me “Ed”, because Ed was my nickname. Then he started a family and moved to Pennsylvania.
Third, I read the entire article. Except for the proofs. And the curriculum at the end. And the second dream. As a student who opted not to take a math class this year (unless physics counts), I agree with a lot of the points made in the article. I find it ironic that despite my accelerated advancement in math (in comparison with peers), one of my best subjects is also the most reliably boring. Apparently I am highly skilled in memorizing formulas and then applying them properly, although I find the process bland and unfulfilling. I have had the incredible fortune of meeting math teachers that engage their students and are passionate about their subject, yet often I need to write a novel in the margins of my notes to stay focused.
On the other hand, that proof with the circle-thing and the triangle-thing scares me to pieces. Yes, it’s beautiful, and therefore it must be beyond my reach. I keep thinking that if math turns into an artistic exploration, then I’ll never learn about limit comparison tests, and then I won’t have learned everything I could have learned.
Then again, I opted out of MVC. And I did think proofs were fun.
If I wanted to make this post even longer, then I could attempt to discuss whether the surrounding culture has pressured me to want to learn about limit comparison tests. Or I could focus on the “therefore it must be beyond my reach” assertion, and that one of my consistent life patterns has been to avoid failure by avoiding the attempt, and then whether that mindset has been cultivated by gifted and talented programs, family, genetics, all of the above, or something else.
And still I would only be focusing on the problem.
I devoutly hope that more people comment, because I feel weird commenting when I’m not in your class.
In the meantime, here is the excerpt that touched me the most:
SIMPLICIO: Yes, but before you can write your own poems you need to learn the alphabet. The process has to begin somewhere. You have to walk before you can run.
SALVIATI: No, you have to have something you want to run toward.
Meredith,
Thanks for visiting the blog. Spread the word–I haven’t really advertised it very widely. And I promise I’ll jazz it up soon!
And thanks for your thoughtful, honest comments. You give a balanced perspective to Lockhart’s essay. As I said in my post, I’m waiting to give my opinion until I hear a few more comments (we’ll see if there are any). I especially appreciated your response to the circle-triangle proof: Yes, the creative process is scary. I mean, how would we ever test your math ability? Maybe we wouldn’t. And yet it’s clear that one of the purposes of traditional math education is to teach you mathematical procedures that produce objective results–results we can easily test. In what ways would Lockhart assess his students? What if they didn’t figure out that circle-triangle proof by the end of the period?
Mr. Chase
Meredith told me about your blog.
I read the article, and I felt kind of scared, at first, because it got me to think about why I liked math class so much (if, like a power-crazed religious institution, the Math Curriculum crushes your spirit and forces you to think of math as rites, procedures). (Run-on sentence there.) I suppose I’ve tried to grasp the beauty of the concepts behind the theorems and formulas, but there isn’t always time for that in class, as Lockhart notes. I feel that more this year, when everything feels like A Comprehensive Guide to A Five on the AP Calculus Exam, despite having an excellent teacher.
As for testing creative thinking: All I can remember from geometry class were the long tests with difficult proofs, on things we weren’t taught (formulaically). In retrospect, I think my geometry teacher was trying to have us think creatively, but it was hard to finish those tests on time. Another reason I’d be afraid: I wouldn’t be able to explore math very well without guidance.
Figuring a proof out always felt good, though. And last year, I liked how we were shown the concepts behind the ambiguous case of the law of sines and the double-angle trig formulas, and proof by induction. I’m quite, overly proud to admit that I’ve written down things like “I think I’m smitten with math” and “math is beautiful” … and, “what if I named my daughter ‘Limaçon’?” in my notes.
But that’s just me, personally. (And I talked too much about myself …) It would be difficult, finding a solution to making math interesting, and beautiful, for everyone without upsetting the educators who want Learning Objectives met.
The article was incredibly refreshing, and articulated the problem that I’ve had with math for years-the focus on the plugging in numbers and getting results, instead of looking at how everything fits together. Up until the seventh grade I could do math, but I didn’t want to. Then I took a geometry class where we never had to memorize the postulates, standard formulas, or vocabulary words. The entire class was a discussion of the theory and logic behind the math, and I loved it. My friends who took the standard geometry class hated it and are thrilled they will never have to take it again, but I occasionally do proofs just for the fun of it.
I think to often math classes get bogged down in endless formulas and memorization and ignore why something is that way, thus souring students on math and denying them a full understanding of the concept. It is important to teach certain formulas and forms in order to test students and because a certain degree of math literacy is required to function in our society, but too often they are thought of as a means to an end, not the end itself.
Standardized test makers and takers have always loved math because there is only one right answer. Perhaps the solution to math’s formula and solution based teaching and assessment is to stop focusing on the answer, pay more attention to the process, and allow it some of ambiguity of english and social studies.
I never realized what else could happen when you take math. Math is essential, however. I must say if we were to do math the “creative way”, it will take even longer to find out teh turth adn what took Pythagoras many years to comlete. We only have 45 minutes of class. I like the current way the best. But in Calc with Apps we apply math to the real world which is good.
This essay greatly reminds me of a quote I once heard- “Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game” [G. H. Hardy]
I can’t tell you how much more interesting all the “unnecessary” portions of math are to me than the actual topics we covered (though this is less the case in HL math).
But having only received math instruction one way in my life, I also have to wonder what math education would be like if we learned math the way he proposed- I definitely feel like what I’ve learned can be helpful in some situations. Lockhart goes a little far, in my opinion- arithmetic, at least, should be retained. I would say that the underlying ideas of Algebra and Calculus are important, but the way they’re taught puts so much emphasis on doing exactly as you’re told. I guess I’m mostly pleased with the result of my math education, but I wish the journey had been more pleasant. [I know you tried, Mr. Chase. I appreciated the balancing cardboard graph.]
Pingback: Learning from Mistakes « Random Walks
Pingback: Teaching math with applications « Random Walks